SL(n) Contravariant Matrix-Valued Valuations on Polytopes

Abstract

All $\textrm{SL}(n)$ contravariant matrix-valued valuations on polytopes in $\mathbb{R}^n$ are completely classified without any continuity assumptions. Moreover, the symmetry assumption of matrices is removed. The general Lutwak-Yang-Zhang matrix turns out to be the only such valuation if $n\geq 4$, while a new function shows up in dimension three. In dimension two, the classification corresponds to the known case of $\textrm{SL}(2)$ equivariant matrix-valued valuations.